Chaos in the unbalance response of a rigid rotor in cavitated squeeze-film dampers without centering springs

Numerical investigation on the unbalance response of a rigid rotor supported by squeeze-film dampers without centering springs revealed some complex bifurcation features that have not been previously reported in the literature. With the variation of the unbalance parameter (U), the period-1 solution was found to undergo a sequence of period-doubling bifurcations that eventually resulted in chaotic motion. The existence of a period-3 solution, which formed a closed bifurcation curve consisting of a pair of saddle nodes, was for the first time observed in such a system. The chaotic attractor arising from the period-doubling cascade of the period-1 solution, which was observed to co-exist with the period-3 attractor in a narrow range of U values, was eventually annihilated in a collision with the unstable period-3 orbit in a boundary crisis. Similar to the bifurcations of the period-1 solution, the period-3 solution was also found to bifurcate into solutions of period-6 and period-12, which eventually led to chaotic motion. A chaotic attractor was also observed to co-exist with a period-4 orbit. The period-4 orbit was found to undergo a sequence of reverse period-doubling bifurcations resulting in a large amplitude period-1 orbit. The occurrence of non-synchronous and chaotic motion in rotating machinery is undesirable and should be avoided as they introduce cyclic stresses in the rotor, which in turn may rapidly induce fatigue failure. The magnitude of rotor unbalance where non-synchronous and chaotic motion were observed in this study, although higher than the permissible unbalance level for rigid rotating machinery, may nevertheless occur with in-service erosion of the rotor or in the event of a partial or an entire blade failure.     

Non-linear dynamic analysis of a flexible rotor supported on porous oil journal bearings

In the present paper, the non-linear dynamic analysis of a flexible rotor with a rigid disk under unbalance excitation mounted on porous oil journal bearings at the two ends is carried out. The system equation of motion is obtained by finite element formulation of Timoshenko beam and the disk. The non-linear oil-film forces are calculated from the solution of the modified Reynolds equation simultaneously with Darcy’s equation. The system equation of motion is then solved by the Wilson-θ method. Bifurcation diagrams, Poincaré maps, time response, journal trajectories, FFT-spectrum, etc. are obtained to study the non-linear dynamics of the rotor-bearing system. The effect of various non-dimensional rotor-bearing parameters on the bifurcation characteristics of the system is studied. It is shown that the system undergoes Hopf bifurcation as the speed increases. Further, slenderness ratio, material properties of the rotor, ratio of disk mass to shaft mass and permeability of the porous bush are shown to have profound effect on the bifurcation characteristics of the rotor-bearing system.     

飞行器内裂纹转子系统的非线性动力学特性研究

在Jeffcott转子的开闭裂纹及方波模型基础上,建立了飞行器内裂纹转子系统的运动模型．数值研究表明:当飞行器以不同的等速度飞行时,转子轴与水平面之间夹角的变化将造成重力分量的变化,从而使转子运动在周期解、拟周期或浑沌状态之间变化,而且出现非线性现象的转速比、刚度变化比等参数的范围、进入和退出浑沌的路径、响应中的频率成份也会发生变化．飞行器的飞行速度变化还会改变裂纹转子响应的稳定性．飞行器等速飞行后的加速过程将引起转子振幅的突升及其后的下降,而且会使裂纹转子系统响应可能在不同的非线性状态下交替改变．     

Transitions to chaos in squeeze-film dampers

This work reports on a numerical study undertaken to investigate the imbalance response of a rigid rotor supported by squeeze-film dampers. Two types of damper configurations were considered, namely, dampers without centering springs, and eccentrically operated dampers with centering springs. For a rotor fitted with squeeze-film dampers without centering springs, the study revealed the existence of three regimes of chaotic motion. The route to chaos in the first regime was attributed to a sequence of period-doubling bifurcations of the period-1 (synchronous) rotor response. A period-3 (one-third subharmonic) rotor whirl orbit, which was born from a saddle-node bifurcation, was found to co-exist with the chaotic attractor. The period-3 orbit was also observed to undergo a sequence of period-doubling bifurcations resulting in chaotic vibrations of the rotor. The route to chaos in the third regime of chaotic rotor response, which occurred immediately after the disappearance of the period-3 orbit due to a saddle-node bifurcation, was attributed to a possible boundary crisis. The transitions to chaotic vibrations in the rotor supported by eccentric squeeze-film dampers with centering springs were via the period-doubling cascade and type 3 intermittency routes. The type 3 intermittency transition to chaos was due to an inverse period-doubling bifurcation of the period-2 (one-half subharmonic) rotor response. The unbalance response of the squeeze-film-damper supported rotor presented in this work leads to unique non-synchronous and chaotic vibration signatures. The latter provide some useful insights into the design and development of fault diagnostic tools for rotating machinery that operate in highly nonlinear regimes.     

Nonlinear dynamic analysis of a hybrid squeeze-film damper-mounted rigid rotor lubricated with couple stress fluid and active control

The hybrid squeeze-film damper bearing with active control is proposed in this paper and the lubricating with couple stress fluid is also taken into consideration. The pressure distribution and the dynamics of a rigid rotor supported by such bearing are studied. A PD (proportional-plus-derivative) controller is used to stabilize the rotor-bearing system. Numerical results show that, due to the nonlinear factors of oil film force, the trajectory of the rotor demonstrates a complex dynamics with rotational speed ratio s. Poincaré maps, bifurcation diagrams, and power spectra are used to analyze the behavior of the rotor trajectory in the horizontal and vertical directions under different operating conditions. The maximum Lyapunov exponent and fractal dimension concepts are used to determine if the system is in a state of chaotic motion. Numerical results show that the maximum Lyapunov exponent of this system is positive and the dimension of the rotor trajectory is fractal at the non-dimensional speed ratio s = 3.0, which indicate that the rotor trajectory is chaotic under such operation condition. In order to avoid the nonsynchronous chaotic vibrations, an increased proportional gain is applied to control this system. It is shown that the rotor trajectory will leave chaotic motion to periodic motion in the steady state under control action. Besides, the rotor dynamic responses of the system will be more stable by using couple stress fluid.     

迟滞型材料阻尼转轴的分岔

应用平均法研究迟滞型材料阻尼转轴的分岔.首先用Hamilton原理推导出复数形式的转轴运动微分方程,然后用平均法求出各阶模态主共振时的平均方程,并分析定常解的稳定性,最后用奇异性理论分析正常运动和失稳运动响应(异步涡动)的分岔.研究表明,一定参数条件下,转轴在通过各阶临界转速(主共振)时,可能会因受到冲击而失稳(Hopf分岔).正常运动响应在不平衡量较大时有滞后和跳跃现象,而失稳运动响应是一类余维数较高的非对称分岔.由于内阻尼的非线性,响应随转速增加时还可能产生二次Hopf分岔,对应原系统的双调幅运动.做好动平衡及提高外阻尼水平是避免这种大幅值自激振动的有效措施.     

Response and bifurcation of rotor with squeeze film damper on elastic support

This paper investigates the nonlinear response and bifurcation of rotor with Squeezed Film Damper (SFD) supported on elastic foundation. The motion equations are derived. To analyze the bifurcation of nonlinear response of SFD rotor, the Floquet Multipliers is obtained by solving the perturbation equations with numerical method. For computing Floquet Multipliers, a novel method is presented in this paper, which can begin integration at the stable solution. Simulation results are given in two figures. One figure, which consists of eight subfigures, gives the effect of rotating speed on the response of SFD damper supported on elastic foundation: with increasing rotating speed, the nonlinear response evolves from quasi-period to period, then jumps between different periods, and finally returns to quasi-period; the corresponding bifurcations are saddle-node bifurcation and secondary Hopf bifurcation. The second figure, which consists of six subfigures, shows that: the support stiffness has large influence on the response of bearings and film force in SFD; large support stiffness can lead to oil whirl in SFD.     

Modeling and analysis of nonlinear rotordynamics due to higher order deformations in bending

A mathematical model incorporating the higher order deformations in bending is developed and analyzed to investigate the nonlinear dynamics of rotors. The rotor system considered for the present work consists of a flexible shaft and a rigid disk. The shaft is modeled as a beam with a circular cross section and the Euler Bernoulli beam theory is applied with added effects such as rotary inertia, gyroscopic effect, higher order large deformations, rotor mass unbalance and dynamic axial force. The kinetic and strain (deformation) energies of the rotor system are derived and the Rayleigh–Ritz method is used to discretize these energy expressions. Hamilton’s principle is then applied to obtain the mathematical model consisting of second order coupled nonlinear differential equations of motion. In order to solve these equations and hence obtain the nonlinear dynamic response of the rotor system, the method of multiple scales is applied. Furthermore, this response is examined for different possible resonant conditions and resonant curves are plotted and discussed. It is concluded that nonlinearity due to higher order deformations significantly affects the dynamic behavior of the rotor system leading to resonant hard spring type curves. It is also observed that variations in the values of different parameters like mass unbalance and shaft diameter greatly influence dynamic response. These influences are also presented graphically and discussed.     

The effects of lateral–torsional coupling on the nonlinear dynamic behavior of a rotating continuous flexible shaft–disk system with rub–impact

This study investigates the lateral–torsional coupling effects on the nonlinear dynamic behavior of a rotating flexible shaft–disk system. The system is modeled as a continuous shaft with a rigid disk in its mid span. Coriolis and centrifugal effects due to shaft flexibility are also included. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed mode method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work include time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The main objective of the present study is to investigate the torsional coupling effects on the chaotic vibration behavior of a system. Periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for cases with and without torsional effects. As demonstrated, inclusion of the torsional–lateral coupling effects can primarily change the speed ratios at which rub–impact occurs. Also, substantial differences are shown to exist in the nonlinear dynamic behavior of the system in the two cases.     

Active vibration control of unbalanced flexible rotor–shaft systems parametrically excited due to base motion

Rotor vibrations caused by large time-varying base motion are of considerable importance as there are a good number of rotors, e.g., the ship and aircraft turbine rotors, which are often subject to excitations, as the rotor base, i.e. the vehicle, undergoes large time varying linear and angular displacements as a result of different maneuvers. Due to such motions of the base, the equations of vibratory motion of a flexible rotor–shaft relative to the base (which forms a non-inertial reference frame) contains terms due to Coriolis effect as well as inertial excitations (generally asynchronous to rotor spin) generated by different system parameters. Such equations of motion are linear but time-varying in nature, invoking the possibility of parametric instability under certain frequency–amplitude combinations of the base motion. An investigation of active vibration control of an unbalanced rotor–shaft system on moving bases is attempted in this work with electromagnetic control force provided by an actuator consisting of four electromagnetic exciters, placed on the stator in a suitable plane around the rotor–shaft. The actuator does not levitate the rotor or facilitate any bearing action, which is provided by the conventional suspension system. The equations of motion of the rotor–shaft continuum are first written with respect to the non-inertial reference frame (the moving base in this case) including the effect of rotor internal damping. A conventional model for the electromagnetic exciter is used. Numerical simulations performed on the flexible rotor–shaft modelled using beam finite elements shows that the control action is successful in avoiding the parametric instability, postponing the instability due to internal material damping and reducing the rotor response relative to the rigid base significantly, with sufficiently low demand of control current in comparison with the bias current in the actuator coils.     

Nonlinear dynamic analysis for bevel-gear system under nonlinear suspension-bifurcation and chaos

This study aims to analyze the dynamic behavior of bevel-geared rotor system supported on a thrust bearing and journal bearings under nonlinear suspension. The dynamic orbits of the system are observed using bifurcation diagrams plotted with both the dimensionless unbalance coefficient and the dimensionless rotational speed ratio as control parameters. The onset of chaotic motion is identified from the phase diagrams, power spectra, Poincaré maps, Lyapunov exponents, and fractal dimensions of the gear-bearing system. The numerical results reveal that the system exhibits a diverse range of periodic, sub-harmonic, and chaotic behaviors. The results presented in this study provide an understanding of the operating conditions under which undesirable dynamic motion takes place in a gear-bearing system and therefore serves as a useful source of reference for engineers in designing and controlling such systems.     

Non-periodic and chaotic response of three-multilobe air bearing system

The dynamic response of a three-multilobe air bearing (TMAB) system is investigated for various values of the rotor mass and bearing number using a hybrid numerical scheme consisting of the differential transformation method (DTM) and the finite difference method (FDM). The validity of the numerical scheme is demonstrated by comparing the results obtained for the rotor center orbit under typical operating conditions with those obtained from the traditional FDM approach and a perturbation method, respectively. The dynamic behavior of the rotor center is then investigated for rotor mass values in the range of 1.0 ≤ m_r ≤ 16.0 kg and bearing number values in the range of 1.0 ≤ Λ ≤ 5.0. The phase trajectories, power spectra, bifurcation diagrams, Poincaré maps and maximum Lyapunov exponents show that the TMAB system exhibits a complex dynamic behavior consisting of periodic, quasi-periodic and chaotic motion at certain values of the rotor mass and bearing number. In general, the numerical results obtained in this study provide a useful insight into the dynamic response of TMAB systems. In particular, the results indicate the operating conditions which should be avoided in order to achieve a desirable periodic motion of the system.     

Nonlinear response of a beam under distributed moving contact load

In this paper, the nonlinear behavior of a one-dimensional model of the disc brake pad is examined. The contact normal force between the disc brake pad lining and rotor is represented by a second order polynomial of the relative displacement between the two elastic bodies. The frictional force due to the sliding motion of the rotor against the stationary pad is modeled as a distributed follower-type axial load with time-dependent terms. By Galerkin discretization, the equation governing the transverse motion of the beam model is reduced to a set of extended Duffing system with quasi-periodically modulated excitations. Retaining the first two vibration modes in the governing equations, frequency response curves are obtained by applying a two-dimensional spectral balance method. For the first time, it is predicted that nonlinearity resulting from the contact mechanics between the disc brake pad lining and rotor can lead to a possible irregular motion (chaotic vibration) of the pad in the neighborhood of simple and parametric resonance. This chaotic behavior is identified and quantitatively measured by examining the Poincaré maps, Fourier spectra, and Lyapunov exponents. It is also found that these chaotic motions emerge as a result of successive Hopf bifurcations characterized by the torus breakdown and torus doubling routes as the excitation frequency varies. Various aspects of the numerical difficulties in the solution of the nonlinear equations are also discussed.     

Steady-state,self-oscillating and chaotic behavior of a PID controlled nonlinear servomechanism by using Bogdanov–Takens and Andronov–Poincaré–Hopf bifurcations

This paper analyzes a controlled servomechanism with feedback and a cubic nonlinearity by means of the Bogdanov–Takens and Andronov–Poincaré–Hopf bifurcations, from which steady-state, self-oscillating and chaotic behaviors will be investigated using the center manifold theorem. The system controller is formed by a Proportional plus Integral plus Derivative action (PID) that allows to stabilize and drive to a prescribed set point a body connected to the shaft of a DC motor. The Bogdanov–Takens bifurcation is analyzed through the second Lyapunov stability method and the harmonic-balance method, whereas the first Lyapunov value is used for the Andronov–Poincaré–Hopf bifurcation. On the basis of the results deduced from the bifurcation analysis, we show a procedure to select the parameters of the PID controller so that an arbitrary steady-state position of the servomechanism can be reached even in presence of noise. We also show how chaotic behavior can be obtained by applying a harmonical external torque to the device in self-oscillating regime. The advantage of achieving chaotic behavior is that it can be used so that the system reaches a set point inside a strange attractor with a small control effort. The analytical calculations have been verified through detailed numerical simulations.     

碰摩转子中弯扭耦合作用的影响分析

建立了一碰摩转子的弯扭耦合数学模型,应用非线性动力学现代理论和一未考虑弯扭耦合的数学模型的动态响应特征进行比较后发现,两类情况下系统都出现了周期、混沌和加周期等非线性动力学现象.虽然两类方程分岔图的变化过程基本相同,但扭转作用的具体影响却不容忽视.这些结果对以后分析转子碰摩现象有一定的参考价值.     

一类含五次非线性恢复力的Duffing系统共振与分岔特性分析

考虑一类含有外激力和五次非线性恢复力的Duffing系统,运用多尺度法求解得到该系统的幅频响应方程,给出不同参数变化下的幅频特性曲线及变化规律,同时利用奇异性理论得到该系统在3种情形下的转迁集及对应的拓扑结构.其次确定系统的不动点,运用Hamilton函数给出该系统的异宿轨,在此基础上,利用Melnikov方法得到该系统在Smale马蹄意义下发生混沌的阈值.而后通过数值仿真给出了系统随外激力、五次非线性项系数变化下的动态分岔与混沌行为,发现存在周期运动、倍周期运动、拟周期运动及混沌等非线性现象.最后运用Lyapunov指数、相轨图和Poincaré截面等非线性方法对理论的正确性进行验证.上述研究结论为进一步提升对Duffing系统非线性特性及其演化规律的认识提供了一定的理论参考.     

Influence of yaw stiffness on the nonlinear dynamics of railway wheelset

A numerical simulation of the dynamic behavior of a railway wheelset is presented. The contact forces between the wheel and the rail are estimated using Johnson and Vermeulen theory of creepages. Nonlinear governing equations of motion of wheelset on a straight track are solved using fourth-order Runge–Kutta method. Both symmetric and asymmetric oscillations and chaotic motion are observed. The influence of yaw stiffness and axial velocity on the response of wheelset is studied. Broadband chaotic motion is developed at various velocity levels. The results are presented in the form of time evolution, phase plots, Poincare maps and bifurcation diagrams. The Lyapunov exponent is calculated and its variation with time is presented. Intermittency is observed. There is a shift in the bifurcation diagram by increasing the yaw stiffness. It indicates that chaotic behavior could be delayed with increasing yaw stiffness.     

Bifurcation and chaos in a discrete time predator-prey system of Leslie type with generalized Holling type III functional response

This paper is devoted to study a discrete time predator-prey system of Leslie type with generalized Holling type III functional response obtained using the forward Euler scheme. Taking the integration step size as the bifurcation parameter and using the center manifold theory and bifurcation theory, it is shown that by varying the parameter the system undergoes flip bifurcation and Neimark-Sacker bifurcation in the interior of $\mathbb{R}_+^2$. Numerical simulations are implemented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as cascade of period-doubling bifurcation in period-$2$, $4$, $8$, quasi-periodic orbits and the chaotic sets. These results shows much richer dynamics of the discrete model compared with the continuous model. The maximum Lyapunov exponent is numerically computed to confirm the complexity of the dynamical behaviors. Moreover, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.     

Nonlinear dynamic analysis of fractional order rub-impact rotor system

Nonlinear dynamic characteristics of rub-impact rotor system with fractional order damping are investigated. The model of rub-impact comprises a radial elastic force and a tangential Coulomb friction force. The fractional order damped rotor system with rubbing malfunction is established. The four order Runge–Kutta method and ten order CFE-Euler method are introduced to simulate the fractional order rub-impact rotor system equations. The effects of the rotating speed ratio, derivative order of damping and mass eccentricity on the system dynamics are investigated using rotor trajectory diagrams, bifurcation diagrams and Poincare map. Various complicated dynamic behaviors and types of routes to chaos are found, including period doubling bifurcation, sudden transition and quasi-periodic from periodic motion to chaos. The analysis results show that the fractional order rub-impact rotor system exhibits rich dynamic behaviors, and that the significant effect of fractional order will contribute to comprehensive understanding of nonlinear dynamics of rub-impact rotor.     

Bifurcation and nonlinear analysis of a flexible rotor supported by a relative short spherical gas bearing system

This paper employs a hybrid numerical method combining the differential transformation method and the finite difference method to study the bifurcation and nonlinear dynamic behavior of a flexible rotor supported by a relative short spherical gas bearing (RSSGB) system. The analytical results reveal a complex dynamic behavior comprising periodic, sub-harmonic, quasi-periodic, and chaotic responses of the rotor center and the journal center. Furthermore, the results reveal the changes which take place in the dynamic behavior of the bearing system as the rotor mass and bearing number are increased. The current analytical results are found to be in good agreement with those of other numerical methods. Therefore, the proposed method provides an effective means of gaining insights into the nonlinear dynamics of RSSGB systems.